Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2)(3,4)(6,10)(8,12)(14,18)(16,19);(6,10)(7,11)(13,14,17,18)(15,19,20,16);(1,2,3,4)(5,6,9,10)(7,8,11,12)(14,18)(16,19);(1,2)(3,4)(6,10)(7,11)(13,15,17,20)(14,19,18,16);(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19);(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72241090, #72241092, #72241121, … |
| maximal normal subgroup of | #72241090, #72241092, #72241121, … |
| maximal quotient of | #72281822, #72329334, #72329447, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589268);